Generalized Exotic Cluster Structures on SL_n for Non-Oriented Belavin-Drinfeld Data
Structures amassées généralisées exotiques sur SL_n pour les données de Belavin-Drinfeld non-orient

18 – 22 May 2020

Cluster algebras were introduced by Fomin and Zelevinsky about twenty years ago and rapidly found their way into the broader mathematical landscape providing universal structures observed in many seemingly unrelated areas of mathematics and mathematical physics. We initiated a systematic study of multiple cluster structures on the rings of regular functions on semi-simple Lie groups following an approach based on the notion of a Poisson bracket compatible with a cluster structure. Examples of such Poisson brackets are provided by Poisson-Lie structures corresponding to quasi-triangular Lie bialgebras. They are associated with solutions of classical Yang-Baxter equation classified by Belavin and Drinfeld in early 80’s. Solutions are described by combinatorial data called Belavin-Drinfeld triples (isometries between two subsets of simple roots) and continuous parameters.
In the case of SLn the Belavin-Drinfeld triple can be described as two collections of disjoint intervals in [1, n1] and an isometry between them. We call the Belavin-Drinfeld triple oriented if the restriction of the isometry to every interval of the first collection preserves the order. Recently, we proved that every oriented Belavin-Drinfeld triple corresponds to (generally speaking, generalized) cluster algebra. The non-oriented case remains largely unexplored. The only existing example of a cluster structure corresponding to a non-oriented triple (forSL5) is built via computer experiments. The goal of our proposed stay at CIRM is to study cluster structures that correspond to non-oriented data. As an immediate application we will obtain an explicit description of quantiz algebras of functions on SLn with respect to all Belavin-Drinfeld Poisson-Lie brackets.

Michael Gekhtman (University of Notre Dame)
Michael Shapiro (Michigan State University)
Alek Vainshtein (University of Haifa)